核實(shí)數(shù)據(jù)下的遞歸核密度估計(jì)

宇世航, 趙世舜

(1. 齊齊哈爾大學(xué) 理學(xué)院, 黑龍江 齊齊哈爾 161006; 2. 吉林大學(xué) 數(shù)學(xué)學(xué)院, 長(zhǎng)春 130012)

基于替代數(shù)據(jù)和核實(shí)樣本推斷的研究目前已有許多結(jié)果[1-10]. Sepanski等[1]研究了基于核實(shí)數(shù)據(jù)的非線性EV模型; Wolvreton等[11]提出了f(x)的遞歸型核密度估計(jì):

由于遞歸型核密度估計(jì)在添加樣本點(diǎn)時(shí), 不必重新計(jì)算所有項(xiàng), 只需計(jì)算添加項(xiàng), 因此使計(jì)算更方便. 基于此, 本文考慮借助于核實(shí)數(shù)據(jù), 構(gòu)造一遞歸型概率密度估計(jì)量, 并研究其漸近正態(tài)性.

1 主要結(jié)果

于是在一些正則條件下,f(x)可被如下遞歸核估計(jì)量一致估計(jì):

定義

AppendixA條件:

(A·f):f(x)是k階有界可導(dǎo)的;

(A·K):K(·)在有界支撐集上是k階非負(fù)有界的核函數(shù);

定理1在AppendixA條件下, 有

證明:

(3)

這里C為任意常數(shù), 且在不同處可取不同的值. 于是, 由式(3)～(6)有

則

從而

其中

而

及條件(A·bn,ηn), 可得

(8)

同理, 有

(10)

于是由式(2)～(10), 有

(11)

同理, 由(A·f),(A·K),(A·hj), 得

(13)

綜上所述, 有

令

由條件(A·K)和(A·h), 有

由式(15)～(17), 顯然有

I1→N(0,θ1σ2(x)),

(18)

I2→N(0,θ2σ2(x)),

(19)

而

故結(jié)合式(15),(18)～(20)可得

2 模擬結(jié)果

(n,N)=(20,100),(50,100),(50,300),(100,300),

圖1 n=20, N=100時(shí)的模擬結(jié)果Fig.1 Simulation result for n=20, N=100

圖2 n=50, N=100時(shí)的模擬結(jié)果Fig.2 Simulation result for n=50, N=100

圖3 n=50, N=300時(shí)的模擬結(jié)果Fig.3 Simulation result for n=50, N=300

圖4 n=100, N=300時(shí)的模擬結(jié)果Fig.4 Simulation result for n=100, N=300

由圖1～圖4可見(jiàn), 給定樣本總數(shù)N的情況下, 模擬效果隨核實(shí)數(shù)據(jù)樣本容量n的增加而漸好； 當(dāng)固定核實(shí)數(shù)據(jù)樣本容量n時(shí), 頂部隨樣本總量N的增加模擬效果漸好, 尾部變差； 如果同時(shí)增大N和n, 模擬結(jié)果更趨近于f(x), 并且也更平滑.

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